MMN-4255

Note on the moment generating function of the multivariate normal distribution

Kenichi Hirose;

Abstract

We present a formula for the derivatives of the moment generating function of the multivariate normal distribution. We formulate it in terms of the summation of the contractions by pairings, which encodes a combinatorial computation procedure. We give two applications. First, we provide a simple proof of Isserlis’ theorem and derive a formula for the moments of the multivariate normal distribution. Second, we prove a formula for the moments of the product of a finite number of correlated normally and lognormally distributed random variables.


Vol. 25 (2024), No. 1, pp. 287-300
DOI: https://doi.org/10.18514/MMN.2024.4255


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